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January 24, 2019

Indian National Math Olympiad, INMO 2019 Problems

This post contains problems from Indian National Mathematics Olympiad, INMO 2019. Try them and share your solution in the comments.

INMO 2019, Problem 1

Let ABC be a triangle with BAC>90. Let D be a point on the line segment BC and E be a point on the line AD such that AB is tangent to the circumcircle of triangle ACD at A and BE perpendicular to AD. Given that CA=CD and AE=CE, determine BCA in degrees

INMO 2019, Problem 2

Let A1B1C1D1E1 be a regular pentagon. For 2n≤11, let AnBnCnDnEn whose vertices are the midpoints of the sides of An−1Bn−1Cn−1Dn−1En−1. All the 5 vertices of each of the 11 pentagons are arbitrarily coloured red or blue. Prove that four points among these 55 have the same colour and form the vertices of a cyclic quadrilateral.

INMO 2019, Problem 3

Let m, n be distinct positive integers. Prove that gcd(m,n)+gcd(m+1,n+1)+gcd(m+2,n+2)≤2|mn|+1. Further, determine when equality holds.

INMO 2019, Problem 4

 Let n and M be positive integers such that M>nn−1. Prove that there are n distinct primes p1, such that pj divides M+j for 1jn.

INMO 2019, Problem 5

Let AB be a diameter of a circle Γ and let C be a point on Γ different from A and B. let D be the foot of perpendicular from C on to AB. let K be a point of the segment CD such that AC is equal to the semiperimeter of the triangle ADK. Show that the excircle of triangle ADK opposite A is tangent to Γ.

INMO 2019, Problem 6

Let \(f\) be a function defined from the set \(\{(x,y) : x,y\) are real, \(xy \neq 0\}\) to the set of all positive real number such that

(i)$f(x y, z)=f(x, z) f(y, z),$ for all $x, y \neq 0$
(ii)$\quad f(x, y z)=f(x, y) f(x, z),$ for all $x, y \neq 0$
(iii)$f(x, 1-x)=1,$ for all $x \neq 0,1$
Prove that
(a) $\quad f(x, x)=f(x,-x)=1,$ for all $x \neq 0$
(b) $f(x, y) f(y, x)=1,$ for all $x, y \neq 0$

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